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Chapter 20: Problem 14

The flux through a loop of wire changes uniformly from \(+40 \mathrm{~Wb}\) to \(-20 \mathrm{~Wb}\) in \(1.5 \mathrm{~ms}\). (a) What is the significance of the negative number attached to the final flux value? (b) What is the average induced emf in the loop? (c) If you wanted to double the average induced emf by changing only the time, what would the new time interval be? (d) If you wanted to double the average induced emf by changing only the final flux value, what would it be?

Short Answer

Expert verified
(a) Negative flux indicates reversed direction. (b) Induced emf is 40000 V. (c) New time interval is 0.75 ms. (d) Final flux should be -80 Wb.

Step by step solution

01

Understanding Flux Changes

In part (a), the negative final flux value indicates that the direction of the magnetic field through the loop has reversed. Magnetic flux (\(\Phi\)) is related to the field's direction with respect to the loop, which means the sign change represents a reversal in the magnetic field's direction.
02

Calculate Average Induced EMF

For part (b), use Faraday's law of electromagnetic induction: \( \text{emf} = -\frac{\Delta \Phi}{\Delta t} \). The change in flux \( \Delta \Phi = -20 \mathrm{~Wb} - (+40 \mathrm{~Wb}) = -60 \mathrm{~Wb} \), and the time interval \( \Delta t = 1.5 \mathrm{~ms} = 0.0015 \mathrm{~s} \). Thus, \( \text{emf} = -\frac{-60 \mathrm{~Wb}}{0.0015 \mathrm{~s}} = 40000 \mathrm{~V} \).
03

Determine New Time Interval for Double EMF

For part (c), if the emf is to be doubled, then \( \text{emf}_{ ext{new}} = 2 \times 40000 \mathrm{~V} = 80000 \mathrm{~V} \). Keeping \( \Delta \Phi \) the same, the new time interval is obtained by \( \text{emf}_{ ext{new}} = -\frac{\Delta \Phi}{\Delta t'} \), or \(\Delta t' = \frac{-60 \mathrm{~Wb}}{80000 \mathrm{~V}} = 0.00075 \mathrm{~s} = 0.75 \mathrm{~ms} \).
04

Changing Final Flux Value to Double EMF

In part (d), to double the emf with the same time interval, the change in flux must double. Thus, \(2 \times 60 \mathrm{~Wb} = 120 \mathrm{~Wb}\). If the initial flux is still \(+40 \mathrm{~Wb}\), then the final flux \( \Phi_f \) should be \( \Phi_f = +40 \mathrm{~Wb} - 120 \mathrm{~Wb} = -80 \mathrm{~Wb} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Flux
Magnetic flux is a measure of the amount of magnetic field passing through a given area. Imagine it like a shower, where the water flow is the magnetic field and the shower base is the area the field passes through. The formula for magnetic flux is given by:\[ \Phi = B \cdot A \cdot \cos(\theta) \]Where:- \( \Phi \) is the magnetic flux.- \( B \) is the magnetic field strength.- \( A \) is the area through which the field lines pass.- \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
A change in the magnetic flux through a loop can occur due to changes in the magnetic field strength, the area of the loop, or the orientation of the loop with respect to the magnetic field. In the given exercise, the flux changes from a positive to a negative value, indicating a reversal in the magnetic field's direction. This change is crucial for understanding electromagnetic induction.
Faraday's Law
Faraday's Law of Electromagnetic Induction is a fundamental principle that explains how a change in magnetic flux can induce an electromotive force (emf) in a closed loop. According to this law:\[ \text{emf} = - \frac{d\Phi}{dt} \]
This means that the emf induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit. The negative sign is a reflection of Lenz's law, which states that the induced emf will create a current that opposes the change in flux.Here are some key points related to Faraday's Law:
  • The faster the magnetic flux changes, the greater the induced emf.
  • Lenz's law ensures that energy conservation is maintained by opposing the change in flux.
  • This principle is applied in various technologies such as transformers and electric generators.

In the context of the exercise, Faraday's Law is used to calculate the average induced emf by considering the change in flux and the time interval over which the change occurs.
Induced EMF
Induced electromotive force (emf) is a voltage generated by the change in magnetic flux through a loop of wire. When the magnetic environment of a coil changes, it "induces" an emf. This principle is the foundation of many practical applications, such as:
  • Electric generators, where mechanical energy is converted into electrical energy.
  • Transformers, which modify voltage levels in power systems.
  • Induction cooktops, which use induced currents to heat pots and pans.

In the exercise, the average induced emf is calculated using the formula derived from Faraday's Law:\[ \text{emf} = -\frac{\Delta \Phi}{\Delta t} \]Where \( \Delta \Phi \) represents the change in magnetic flux and \( \Delta t \) is the time interval over which this change occurs. To determine the average induced emf:- Calculate the total change in flux.- Divide by the time interval.
This exercise also explores the effects of adjusting the flux value or time interval to achieve a different emf, illustrating the versatile application of these concepts in real-world scenarios.

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Most popular questions from this chapter

A solenoid of length \(40.0 \mathrm{~cm}\) is made of 10000 circular coils. It carries a steady current of 12.0 A. Near its center is placed a small, flat, circular metallic coil of 200 circular loops, each with a radius of \(2.00 \mathrm{~mm}\). This small coil is oriented so that it receives half of the maximum magnetic flux. A switch is opened in the solenoid circuit and its current drops to zero in \(25.0 \mathrm{~ms}\). (a) What was the initial flux through the small coil? (b) Determine the average induced emf in the small coil during the \(25.0 \mathrm{~ms}\). (c) If you look along the long axis of the solenoid so that the initial 12.0 A current is clockwise, determine the direction of the induced current in the small inner coil during the time the current drops to zero. (d) During the \(25.0 \mathrm{~ms}\), what was the average current in the small coil, assuming it has a resistance of \(0.15 \Omega ?\)

Assume that a uniform magnetic field exists perpendicular to the plane of this page (into it) and has a strength of \(0.150 \mathrm{~T}\). Assume further that this field ends sharply at the paper's edges. A single circular loop of wire is also in the plane of the paper and moves across it from left to right at a speed of \(1.00 \mathrm{~m} / \mathrm{s}\). The loop has a radius of \(1.50 \mathrm{~cm}\). The loop starts with its center \(10.0 \mathrm{~cm}\) to the left of the left edge, in zero field, enters the field, then exits at the right edge back into zero field until its center is \(10.0 \mathrm{~cm}\) to the right of the right edge. (a) Make a sketch of the induced emf in the coil versus time, putting numbers on the time axis and taking positive emf to indicate clockwise direction and negative emf to indicate counterclockwise (the emf axis will not have any numbers on it.) (b) What is the average emf (magnitude) induced in the coil when it is (1) to the left of the left edge, (2) entering the left side of the field, (3) completely in the field region, (4) exiting the right field edge, and (5) out in the zero field region to the right of the right edge.

Microwave ovens can have cold spots and hot spots due to standing electromagnetic waves, analogous to standing wave nodes and antinodes in strings (v Fig. 20.32). (a) The longer the distance between the cold spots, (1) the higher the frequency of the waves, (2) the lower the frequency of the waves, (3) the frequency of the waves is independent of this distance. Why? (b) In your microwave the cold spots (nodes) occur approximately every \(5.0 \mathrm{~cm},\) but your neighbor's microwave produces them at every \(6.0 \mathrm{~cm}\). Which microwave operates at a higher frequency and by how much?

An ideal solenoid with a current of 1.5 A has a radius of \(3.0 \mathrm{~cm}\) and a turn density of 250 turns \(/ \mathrm{m}\). (a) What is the magnetic flux (due to its own field) through only one of its loops at its center? (b) What current would be required to double the flux value in part (a)?

A metal airplane with a wingspan of \(30 \mathrm{~m}\) flies horizontally along a north-south route in the northern hemisphere at a constant speed of \(320 \mathrm{~km} / \mathrm{h}\) in a region where the vertical component of the Earth's magnetic field is \(5.0 \times 10^{-5} \mathrm{~T}\). (a) What is the magnitude of the induced emf between the tips of its wings? (b) If the easternmost wing tip is negatively charged, is the plane flying due north or due south? Explain.
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